Lecture Notes

• Lecture 0 (01/08) Definitions: Categories. Morphisms - injective, surjective, bijective, left/right inverse, isomorphisms. Objects - sub/quotient, initial/final, null.
• Lecture 1 (01/10) Functors. Natural transformations. Equivalence of categories.
• Lecture 2 (01/12) Equivalence of categories cntd. Dual categories. Product of catgories.

There is a mistake in Lecture 2. See section 3.0 of the next lecture for errata

• Lecture 3 (01/17) Adjoint functors. Unit and counit of adjunction.
• Lecture 4 (01/19) Adjoint functors cntd. Yoneda's lemma. Representable functors.
• Lecture 5 (01/22) (Functors defining objects). Direct sums and direct products.
• Lecture 6 (01/24) (Functors defining objects). Inverse limit.
• Lecture 7 (01/26) Direct and inverse limits cntd.
• Lecture 8 (01/29) Additive categories. Finite direct sums/products. Kernel and cokernel.
• Lecture 9 (01/31) Kernel and cokernel continued. Abelian categories. Additive functors.
• Lecture 10 (02/02) Notion of exactness: of sequences; of functors. Left/right exact functors. Hom functor.

• Lecture 11 (02/05) Long road ahead - towards derived functors. To do list.
• Lecture 12 (02/07) Injective and projective objects - "effacability".
• Lecture 13 (02/09) Tensor functor. Right exactness.
• Lecture 14 (02/12) Category of complexes. Cohomology functors. Snake lemma.
• Lecture 15 (02/14) Long exact sequence in cohomology. Homotopy.
• Lecture 16 (02/16) Uniqueness (up to homotopy) of injective and projective resolutions.
• Lecture 17 (02/19) Baer's criterion of injectivity. Examples of injective abelian groups. Existence of injective R-modules.
• Lecture 18 (02/21) Cogenerators. Injective envelops. Existence of injective/projective resolutions.
• Lecture 19 (02/23) Two ways of defining Ext. Result that they are the same.
• Lecture 20 (02/26) Ext vs extensions. Some examples.
• Lecture 21 (02/28) Group cohomology - a glimpse into group extensions.
• Lecture 22 (03/02) Flat modules. Torsion-free modules.
• Lecture 23 (03/05) Flat modules continued. Flat (assuming some finiteness conditions) over a local ring is free.
• Lecture 24 (03/07) Definition of Tor functors. Some computations.
• Lecture 25 (03/09) Brief summary of Ext and Tor (analogy with cohomology and homology).

• Lecture 26 (03/19) Field extensions. Degree of an extension. Algebraic vs transcendental elements.
• Lecture 27 (03/21) A little review. Splitting field of a polynomial - existence.
• Lecture 28 (03/23) Splitting field - uniqueness. Independence of group characters. Lower bound on degree of an extension.
• Lecture 29 (03/26) Artin's theorem. Galois extensions. Normal and separable extensions. Galois iff normal and separable (finite case).
• Lecture 30 (03/28) Splitting field of a set of polynomials - existence and uniqueness. Algebraically closed fields.
• Lecture 31 (03/30) Some properties of normal extensions. Galois iff normal and separable (general case).
• Lecture 32 (04/02) Fundamental theorem of Galois theory (finite case).
• Lecture 33 (04/04) Differential criterion of separability. Finite fields. Perfect/imperfect fields.
• Lecture 34 (04/06) Primitive element theorem I.
• Lecture 35 (04/09) Primitive element theorem II. Normal basis of Galois extensions.
• Lecture 36 (04/11) Roots of unity. Noether's equations. A little bit of Galois cohomology.
• Lecture 37 (04/13) Kummer theory.
  Errata. Corollary on page 7 should have the condition that sigma is not identity.
• Lecture 38 (04/16) Norm, trace, Hilbert 90 and application to cyclic extensions.
• Lecture 39 (04/18) Topological groups. Profinite groups.
• Lecture 40 (04/20) Fundamental theorem of (infinite) Galois extensions.

Optional reading material

• Appendix A Grothendieck's axioms (AB3-5) and proof of Baer's criterion of injectivity